 1.3.1: For Exercises 18, draw and carefully label the figures. Use the app...
 1.3.2: For Exercises 18, draw and carefully label the figures. Use the app...
 1.3.3: For Exercises 18, draw and carefully label the figures. Use the app...
 1.3.4: For Exercises 18, draw and carefully label the figures. Use the app...
 1.3.5: For Exercises 18, draw and carefully label the figures. Use the app...
 1.3.6: For Exercises 18, draw and carefully label the figures. Use the app...
 1.3.7: For Exercises 18, draw and carefully label the figures. Use the app...
 1.3.8: For Exercises 18, draw and carefully label the figures. Use the app...
 1.3.9: Which creatures in the last group below are Zoids? What makes a Zoi...
 1.3.10: What are the characteristics of a good definition?
 1.3.11: What is the difference between complementary and supplementary angles?
 1.3.12: If X and Y are supplementary angles, are they necessarily a linear ...
 1.3.13: Write these definitions using the classify and differentiate method...
 1.3.14: There is something wrong with this definition for a pair of vertica...
 1.3.15: For Exercises 1524, four of the statements are true. Make a sketch ...
 1.3.16: For Exercises 1524, four of the statements are true. Make a sketch ...
 1.3.17: For Exercises 1524, four of the statements are true. Make a sketch ...
 1.3.18: For Exercises 1524, four of the statements are true. Make a sketch ...
 1.3.19: For Exercises 1524, four of the statements are true. Make a sketch ...
 1.3.20: For Exercises 1524, four of the statements are true. Make a sketch ...
 1.3.21: For Exercises 1524, four of the statements are true. Make a sketch ...
 1.3.22: For Exercises 1524, four of the statements are true. Make a sketch ...
 1.3.23: For Exercises 1524, four of the statements are true. Make a sketch ...
 1.3.24: For Exercises 1524, four of the statements are true. Make a sketch ...
 1.3.25: For Exercises 25 and 26, refer to the graph at right. Find possible...
 1.3.26: For Exercises 25 and 26, refer to the graph at right. Find possible...
 1.3.27: A partial mirror reflects some light and lets the rest of the light...
 1.3.28: Find possible coordinates of points A, B, and C on the graph at rig...
 1.3.29: If D is the midpoint of AC and C is the midpoint of AB , and AD = 3...
 1.3.30: If BD is the angle bisector of ABC, BE is the angle bisector of ABD...
 1.3.31: Draw and label a figure that has two congruent segments and three c...
 1.3.32: Show how three lines in a plane can have zero, exactly one, exactly...
 1.3.33: Show how it is possible for two triangles to intersect in one point...
 1.3.34: Each pizza is cut into slices from the center. a. What fraction of ...
Solutions for Chapter 1.3: Whats a Widget?
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 1.3: Whats a Widget?
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4. Chapter 1.3: Whats a Widget? includes 34 full stepbystep solutions. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. Since 34 problems in chapter 1.3: Whats a Widget? have been answered, more than 22110 students have viewed full stepbystep solutions from this chapter.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.